A systematic theory of quantum fluctuation in the time-dependent linearly driven parametric amplification is developed. At first, a time-dependent linearly driven Fokker-Planck equation for the degenerate optical parametric amplification system is deduced when the pump depletion is considered, and then the quantum fluctuation below or near the threshold is evaluated, which is in agreement with that obtained by the linear theory or perturbation expansion near the threshold. Above the threshold, the short-time behavior of our solution is close to the linearization approximation; however, with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is quite different from the linear theory.
A systematic theory of quantum fluctuation in the time-dependent linearly driven parametric amplification is developed. At first, a time-dependent linearly driven Fokker-Planck equation for the degenerate optical parametric amplification system is deduced when the pump depletion is considered, and then the quantum fluctuation below or near the threshold is evaluated, which is in agreement with that obtained by the linear theory or perturbation expansion near the threshold. Above the threshold, the short-time behavior of our solution is close to the linearization approximation; however, with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is quite different from the linear theory.
The time-dependent, linearly driven Fokker-Planck equation for the degenerate parametric amplifier is derived when the pump depletion is present both above and below threshold. A model that the driving force εα1 is a product of the classical quantity ε and quantum operator α1 is proposed. It is found that below or near threshold, our results conform to the linear theory or perturbation series expansion, and above threshold, the short-time behavior of our solution is close to the linearization approximation; with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is always quite different from the linear theory. It is shown that the maximal squeezing can be obtained at large pump parameter μ=ε/k and interaction time Δτ.
An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).
In this paper, we first find out the analytic solution of the time-dependent Fokker-Planck equation of the non-degenerate optical parametric amplification (NOPA) system under the consideration of the dispersion, the loss and the pump depletion effects. Then, through the numerical calculation, we obtain the squeezing characteristic of the degenerate optical parametric amplification (DOPA) system with dispersion. the research indicates: the dispersion effect stems from the nonlinear susceptibility change from χ″ toχ″/{1+σ2/}/+2, with the increasing of the dispersion effect, the general feature of the squeezing curves beeps unchanged, and the curves contract toward left. The maximum squeezing approaches to the linear theory 1/(1+μ). Finally, we obtain the entanglement characteristic of the NOPA system with dispersion. We find out when σ is given, with the increasing of pump parameter μ, he corresponding phase makes a large change. The nonlinear susceptibility changes many times. When the polarity is positive, the system obtains the gain, when the polarity is negative, the system suffers the loss, but the gain is mainly dissipated by the loss, so the net gain is small, the squeezing is also small. The minimum variance V1 reduces gradually, and the whole curve moves to the right, approaches to the linear theory 1/(1+μ).
The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator. The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.
The solution of the time-dependent periodic pumping non-degenerate optical parametric amplifier (NOPA) is derived when the pump depletion is considered both above and below thresholds. Based on this solution, the quantum fluctuation calculated shows that a high entanglement and a good squeezing degree of the parametric light beams are achieved near and above thresholds. We adopt two kinds of pump fields: (i) a continuously modulated pump with a sinusoidal envelope; (ii) a sequence of laser pulses with Gaussian envelopes. We analyse the time evolution of continuous variable entanglement by analytical and numerical calculations, and show that the periodic driven pumping also improves the degree of entanglement. The squeezing and Einstein–Podolsky–Rosen (EPR) entanglement by using the two pumping driven functions are investigated from below to above the threshold regions, the tendencies are nearly the same, and the Gaussian driven function is superior to that of the sine driven function, when the maximum squeezing and the minimum variance of quantum fluctuation are considered. In the meantime, the generalization of frequency degenerate OPA to frequency non-degenerated OPA problem is investigated.
A quantum-statistical treatment of degenerate (single-mode) parametric amplification is given in which the pump is allowed to have time-dependent amplitude and phase. This relaxes earlier restrictions of a constant pump in exact resonance with the signal. Solutions are obtained for the case where the ratio of the time derivative of the pump phase to the coupling strength between pump and signal is constant. This includes the special case of simple offresonance, where the pump has a constant frequency differing from twice the signal frequency by a given amount (and has constant amplitude). The Wigner distribution for the signal mode is obtained (the P distribution does not in general exist as a well-defined function beyond a finite time interval for this process) and is used to determine the time development of the expectation values of various field operators of interest, such as field strengths and fluctuations. A critical point exists in the solution for the photon annihilation operator such that the behavior of the solution and the resulting quantities which it determines is divided into an exponential region and an oscillatory region. The critical point defines, from the expectation value for the photon number, the condition for amplification. For the case of simple off-resonance, this gives the frequency range of amplification. A calculation without using the rotating-wave approximation is given in the Appendix which shows that, to first order, the same range is obtained as when the approximation is used. The effect of damping is given, showing, e.g., how it slows the growth rate of the signal (in the region of amplification) and how it narrows the frequency range of amplification. The effect of a time-dependent pump (and damping) on the breakdown of the P distribution is also described.